Decomposition rank and $\mathcal{Z}$ -stability

Abstract

We show that separable, simple, nonelementary, unital C*-algebras with finite decomposition rank absorb the Jiang–Su algebra $\mathcal{Z}$ tensorially. This has a number of consequences for Elliott’s program to classify nuclear C*-algebras by their K-theory data. In particular, it completes the classification of C*-algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K-groups.